To solve this problem, let's carefully analyze the diagram:

Observations:

1. The diagram has two right triangles, namely:

   • $\triangle XYZ$ (the larger triangle)
   • $\triangle WYX$ (the smaller triangle within the larger triangle)

2. Both triangles share the angle at $Y$, and both have a right angle ($90^\circ$). By the AA similarity criterion (Angle-Angle), the two triangles are similar: $\triangle XYZ \sim \triangle WYX$

Using Similarity:

Since the triangles are similar, their corresponding sides are proportional: $\frac{\text{Side in } \triangle XYZ}{\text{Corresponding side in } \triangle WYX} = \text{constant ratio}.$

Given Values:

From the diagram:

• $YZ = 4$, $ZX = 12$, $XY = a$ in the larger triangle ($\triangle XYZ$).
• $WY = 6$, $WX = b$, $YX = 9$ in the smaller triangle ($\triangle WYX$).

Setting Proportions:

From the similarity of triangles, we can set up the following relationships:

1. $\frac{YZ}{WY} = \frac{ZX}{WX} = \frac{XY}{YX}$.

Using the given values:

1. $\frac{YZ}{WY} = \frac{4}{6} = \frac{2}{3}$,
2. $\frac{ZX}{WX} = \frac{12}{b}$,
3. $\frac{XY}{YX} = \frac{a}{9}$.

Solving for Missing Sides:

1. From $\frac{ZX}{WX} = \frac{2}{3}$: $\frac{12}{b} = \frac{2}{3} \implies b = \frac{12 \cdot 3}{2} = 18.$

2. From $\frac{XY}{YX} = \frac{2}{3}$: $\frac{a}{9} = \frac{2}{3} \implies a = \frac{9 \cdot 2}{3} = 6.$

Final Answer:

• The similar triangles are: $\triangle XYZ \sim \triangle WYX$.
• Missing sides:
  • $a = 6$,
  • $b = 18$.

Would you like me to explain the proportions in more detail or apply another method?

Here are 5 related questions to expand your understanding:

1. How do we determine similarity in triangles using the AA criterion?
2. What other criteria exist for determining triangle similarity (e.g., SSS, SAS)?
3. How can we apply the Pythagorean theorem to verify triangle dimensions?
4. Why is the ratio of corresponding sides constant for similar triangles?
5. How does this method extend to non-right triangles?

Tip: When solving problems with similar triangles, always identify corresponding angles and sides carefully to set up correct proportions.